Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementarymathematics: the ratio of two heights, for example, is a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity.

In this paper I examine Aquinas’s commentary on a text of Aristotle in which the type of order between substance and accidents is discussed. I claim that Aquinas maintains that there cannot be any reference to sensibility, despite any prima facie interpretation of Aristotle’s texts, according to which it could be thought that substance is temporally prior to accidents and, hence, that we must presuppose a perceivable change in the world on the basis of which it is possible to consider something temporally prior to something else. This interpretation – which is possible on the basis of Aristotle’s texts – would be a misinterpretation, according to Aquinas. Aquinas’s assumption is philosophically worthwhile because it confi rms that every metaphysical proposition must abstract from sensibility.

The present paper deals with the notion of essence in Thomas Aquinas. Part 1 focuses on the main points of Aquinas’s doctrine of essence set out in his De ente et essentia, and especially on the concept of essence as the “form of the whole” and the concept of the “nature considered absolutely”. The comparison with the teaching of Aristotle and Aquinas’s Arabic predecessors in Part 2 shows that Thomas’s notion of essence is an innovative re-interpretation, which he largely owes to Avicenna, of the original Peripatetic doctrine. Nevertheless, it is shown that this re-interpretation is to be understood as a result of Avicenna’s and Aquinas’s effort to provide a consistent explanation of various statements in Aristotle’s writings, not always compatible with each other.

Aristotle’s subtle distinction between the forms of friendship and his concept of loving friend as one’s other self propose a solution to the fundamental objection to any eudaimonian theory of slavery, namely that friendship – as basically non-moral phenomenon – is but an egoistic device of one’s happy life. Aristotelian theorems are based on his concept of analogy and on a philosophically specific notion of “self”. Since both of these are rooted in Platonism, Aristotle has toevolve them dialectically in a critical distance to Plato. Still, his dialectical theory of friendship needs to be rooted not in metaphysics but in political theory after all. Political friendship as a utopian perspective taken by each of the citizens in their pursuit of a close relationship with any other indicates a notion of “infinity as perfection” which presents the decisive step beyond Plato and toward the later course of the history of philosophy.